Great Theoretical Ideas In Computer Science John Lafferty CS 15 251 Lecture 25 November 21 2006 Fall 2006 Carnegie Mellon University Cantor s Legacy Infinity And Diagonalization Ideas from the course Induction Numbers Representation Finite Counting and Probability A hint of the infinite Infinite row of dominoes Infinite sums formal power series Infinite choice trees and infinite probability Infinite RAM Model Platonic Version One memory location for each natural number 0 1 2 Aristotelian Version Whenever you run out of memory the computer contacts the factory A maintenance person is flown by helicopter and attaches 100 Gig of RAM and all programs resume their computations as if they had never been interrupted The Ideal Computer no bound on amount of memory no bound on amount of time Ideal Computer is defined as a computer with infinite RAM You can run a Java program and never have any overflow or out of memory errors An Ideal Computer It can be programmed to print out 2 2 0000000000000000000000 1 3 0 33333333333333333333 1 6180339887498948482045 e 2 7182818284559045235336 3 14159265358979323846264 Printing Out An Infinite Sequence A program P prints out the infinite sequence s0 s1 s2 sk if when P is executed on an ideal computer it outputs a sequence of symbols such that The kth symbol that it outputs is sk For every k2 P eventually outputs the kth symbol I e the delay between symbol k and symbol k 1 is not infinite Computable Real Numbers A real number R is computable if there is a program that prints out the decimal representation of R from left to right Thus each digit of R will eventually be output Are all real numbers computable Describable Numbers A real number R is describable if it can be denoted unambiguously by a finite piece of English text 2 Two The area of a circle of radius one Are all real numbers describable Is every computable real number also a describable real number And what about the other way Computable R some program outputs R Describable R some sentence denotes R Computable describable Theorem Every computable real is also describable Computable describable Theorem Every computable real is also describable Proof Let R be a computable real that is output by a program P The following is an unambiguous description of R The real number output by the following program P MORAL A computer program can be viewed as a description of its output Syntax The text of the program Semantics The real number output by P Are all reals describable Are all reals computable We saw that computable describable but do we also have describable computable Questions we will answer in this and next lecture Correspondence Principle If two finite sets can be placed into 1 1 onto correspondence then they have the same size Correspondence Definition In fact we can use the correspondence as the definition Two finite sets are defined to have the same size if and only if they can be placed into 1 1 onto correspondence Georg Cantor 1845 1918 Cantor s Definition 1874 Two sets are defined to have the same size if and only if they can be placed into 1 1 onto correspondence Cantor s Definition 1874 Two sets are defined to have the same cardinality if and only if they can be placed into 1 1 onto correspondence Do and have the same cardinality 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 The even natural numbers and do not have the same cardinality is a proper subset of with plenty left over The attempted correspondence f x x does not take onto and do have the same cardinality 0 1 2 3 4 5 0 2 4 6 8 10 f x 2x is 1 1 onto Lesson Cantor s definition only requires that some 1 1 correspondence between the two sets is onto not that all 1 1 correspondences are onto This distinction never arises when the sets are finite Cantor s Definition 1874 Two sets are defined to have the same size if and only if they can be placed into 1 1 onto correspondence You just have to get used to this slight subtlety in order to argue about infinite sets Do and have the same cardinality 0 1 2 3 4 5 6 7 2 1 0 1 2 3 No way is infinite in two ways from 0 to positive infinity and from 0 to negative infinity Therefore there are far more integers than naturals Actually no and do have the same cardinality 0 1 2 3 4 5 6 0 1 1 2 2 3 3 f x x 2 if x is odd x 2 if x is even Transitivity Lemma Transitivity Lemma Lemma If f A B is 1 1 onto and g B C is 1 1 onto Then h x g f x defines a function h A C that is 1 1 onto Hence and all have the same cardinality Do and have the same cardinality 0 1 2 3 4 5 6 7 The Rational Numbers No way The rationals are dense between any two there is a third You can t list them one by one without leaving out an infinite number of them Don t jump to conclusions There is a clever way to list the rationals one at a time without missing a single one First let s warm up with another interesting example can be paired with Theorem and have the same cardinality Theorem and have the same cardinality 4 3 The point x y represents the ordered pair x y 2 1 0 0 1 2 3 4 Theorem and have the same cardinality 4 6 3 2 3 1 1 0 0 0 The point x y represents the ordered pair x y 7 4 8 5 2 1 2 9 3 4 Defining 1 1 onto f let i 0 will range over N for sum 0 to forever generate all pairs with this sum for x 0 to sum y sum x define f i the point x y i Onto the Rationals The point at x y represents x y 3 0 1 2 The point at x y represents x y Cantor s 1877 letter to Dedekind I see it but I don t believe it Countable Sets We call a set countable if it can be placed into 1 1 onto correspondence with the natural numbers Hence and are all countable Do and have the same cardinality 0 1 2 3 4 5 6 7 The Real Numbers No way You will run out of natural numbers long before you match up every real Now hang on a minute You can t be sure that there isn t some clever correspondence that you haven t thought of yet I am sure Cantor proved it To do this he invented a very important technique called Diagonalization Theorem The set 0 1 of reals between 0 and 1 is not countable Proof by contradiction Suppose 0 1 is countable Let f be a 1 1 onto function from to …
View Full Document
Unlocking...